Mark Cubbon took to Twitter after posts by a woman on Facebook showed photographs inside the Cosham hospital, claiming it was the ‘quietest I have ever seen it’.
It's understood the incident is being reported to Hampshire police by those who have seen the post today.
Mr Cubbon, chief executive of Portsmouth Hospitals University NHS Trust, said Covid patients were increasing at the site.
He tweeted: ‘Saddened to see messages on social media suggesting our wards are empty and pressures we say we’re facing are fictional.
‘We’re treating (increasing numbers of patients) with Covic-19.
‘Our staff are doing an incredible job under a great deal of pressure. Please don’t disrespect them.’
Professor Partha Kar backed Mr Cubbon’s post and added: ‘We have – as of tomorrow – rheumatologists, dermatologists, endocrinologists cancelling clinics to help on general & surgical wards to cover for medical patients.
‘Due to extra pressure on all other teams. This is not fictional.’
The Facebook posts, by Hannah Dean, are largely of corridors.
They do not show the restricted access Covid wards, which are kept separate from general access to prevent infections spreading.
Writing in her post she said: ‘Hospital is the quietest I have ever seen it! I walked all over the hospital, including A&E!
‘I know this is hard to get our heads around, but the government are lying to us! And the reason why they’re lying to us..... is very disturbing!’
Her false claims come as medics are treating 342 patients with Covid-19 at Queen Alexandra Hospital in Portsmouth – with a third of general beds occupied by those with the infection, latest figures show.
Figures from NHS England show that as of 8am on December 29 there were 342 beds occupied by patients with Covid-19 at QA Hospital, up from 134 on November 29. A month earlier than that there were just 35.
Around a third of general and acute beds are occupied by Covid patients.
The figures show 288 adult general and acute beds at QA were occupied with Covid-19 patients as of December 29. There were 596 occupied by non-Covid patients, and 16 empty in the hospital.
As reported, medical director Dr John Knighton urged people to ‘seriously think’ about their actions and make sure they ‘minimise all contact with others’.
On Sunday he tweeted: ‘Let nobody be in any doubt just how hard colleagues and friends across (QA Hospital) and wider (NHS) are working now, for many weeks and for many to come.
‘Shame on those wilfully pretending otherwise or ignoring guidance & tier four rules.’
Previously he said: ‘We are in a period where numbers of cases of Covid are rising rapidly in our hospital and we know they're going to continue to rise for some time until the tier four restrictions starts to have an effect, hopefully.
‘Our staff have been working flat out across many areas of the hospital. Most areas of the hospital now are working way beyond their normal capacity.’
He added: ‘We all have a role in protecting our NHS for those who really need it.’
The number of patients on mechanical ventilation beds has also increased, with 24 as of December 29, up from 15 the month before.
The total number of Covid patients admitted to QA Hospital, and those who were diagnosed in hospital, was 50 in the 24 hours to December 27. This is the latest figure.
On Saturday, Dr Rachel Clarke, a palliative care medic and author, tweeted: ‘Critically ill Covid patients in Kent are being transported by ambulance and helicopter as far as Bristol, Leeds, Southampton & Portsmouth.’
More than 420 people have died at QA Hospital with Covid-19.
Total Covid patients in QA Hospital
December 29: 342
December 28: 352
December 27: 342
December 26: 323
December 25: 300
December 24: 297
December 23: 303